Noncommutative Invariants of Singularities and Application to Index Theory

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

Abstract Award: DMS 1105670, Principal Investigator: Markus J. Pflaum The proposed work will advance the study of singularities by means of noncommutative geometry. Spaces with singularities appear abundantly and naturally in various areas of mathematics. Standard methods developed to study smooth manifolds or smooth varieties can in general not be extended to the singular setting, so one has to develop new approaches. Among the most promising new and original proposals which will provide progress for singularity theory is the idea to determine the cyclic homology theory of function algebras over spaces with singularities. This is the viewpoint from noncommutative geometry which goes back to the work of A. Connes and which has turned out to provide deeper mathematical insight not only into the structure theory of noncommutative but also of commutative algebras. In addition to the computation of cyclic homologies of function algebras over singular spaces, the PI plans to combine recent results from the stratification theory of singular spaces with noncommutative geometry to open up new paths to examine singularities. The construction of new topological invariants of singularities by this approach also promises to provide progress for index theory over spaces with singularities. In particular, it is intended to define inertia spaces associated to proper Lie groupoids and study their singularity structure with the goal of constructing a mathematical device which keeps track of the contribution of singularities to the cyclic homology of convolution algebras over proper Lie groupoids. Finally, relative cyclic cohomology theory will be used to construct and describe secondary invariants of geometric operators in singular situations. Singularity theory is the mathematical discipline in which one describes and studies geometrical objects containing so-called singularities such as corners, edges or vertices. Besides these rather elementary singularities, considerably more complicated ones appear not only in mathematics itself but also in many physical or technical applications like for example hydro dynamics, string theory, robotics or catastrophe theory, which plays a fundamental role in the theoretical understanding of "catastrophic" phenomena in laser physics or population dynamics. A better mathematical understanding of singularities therefore will not only lead to progress within mathematics but also will have its impact for theoretical physics or engineering in situations where singular phenomena appear. The proposed project aims at improving the foundational knowledge on singularities by connecting singularity theory to another modern mathematical theory, namely noncommutative geometry. It is to be expected that this way new mathematical invariants for singularities can be constructed. This will provide further crucial steps towards a classification of singularities as they appear in mathematics, the sciences or engineering. To strengthen the broader impact of the project, the PI plans to present visualizations of singularities via a website specifically designed to disseminate mathematical knowledge.